In Java I might do:
public static void doSomething();
And then I can access the method statically without making an instance:
c         
        I am using ruby 1.9.3 and the program is running smoothly in my irb as well.
1.9.3-p286 :001 > class Ask
1.9.3-p286 :002?>     def self.make_permalink(phrase)
1.9.3-p286 :003?>         phrase.strip.downcase.gsub! /\ +/, '-'
1.9.3-p286 :004?>       end
1.9.3-p286 :005?>   end
 => nil 
1.9.3-p286 :006 > Ask.make_permalink("make a slug out of this line")
 => "make-a-slug-out-of-this-line"
It's also working in my test script. Nothing wrong with your given code.It's fine.
There is one more syntax which is has the benefit that you can add more static methods
class TestClass
  # all methods in this block are static
  class << self
    def first_method
      # body omitted
    end
    def second_method_etc
      # body omitted
    end
  end
  # more typing because of the self. but much clear that the method is static
  def self.first_method
    # body omitted
  end
  def self.second_method_etc
    # body omitted
  end
end
Here's my copy/paste of your code into IRB. Seems to work fine.
$ irb
1.8.7 :001 > class Ask
1.8.7 :002?>   
1.8.7 :003 >   def self.make_permalink(phrase)
1.8.7 :004?>     phrase.strip.downcase.gsub! /\ +/, '-'
1.8.7 :005?>   end
1.8.7 :006?>   
1.8.7 :007 > end
 => nil 
1.8.7 :008 > Ask.make_permalink("make a slug out of this line")
 => "make-a-slug-out-of-this-line"
Seems to work. Test it out in your irb as well, and see what results you're getting. I'm using 1.8.7 in this example, but I also tried it in a Ruby 1.9.3 session and it worked identically.
Are you using MRI as your Ruby implementation (not that I think that should make a difference in this case)?
In irb make a call to Ask.public_methods and make sure your method name is in the list. For example:
1.8.7 :008 > Ask.public_methods
 => [:make_permalink, :allocate, :new, :superclass, :freeze, :===, 
     ...etc, etc.] 
Since you also marked this as a ruby-on-rails question, if you want to troubleshoot the actual model in your app, you can of course use the rails console: (bundle exec rails c) and verify the publicness of the method in question.
Your given example is working very well
class Ask
  def self.make_permalink(phrase)
    phrase.strip.downcase.gsub! /\ +/, '-'
  end
end
Ask.make_permalink("make a slug out of this line")
I tried in 1.8.7 and also in 1.9.3 Do you have a typo in you original script?
All the best